Johann Friedrich Pfaff
Johann Friedrich Pfaff was a German mathematician. He was described as one of Germany's most eminent mathematicians during the 19th century, he was a precursor of the German school of mathematical thinking, which under Carl Friedrich Gauss and his followers determined the lines on which mathematics developed during the nineteenth century. He received his early education at the Carlsschule, where he met Friedrich Schiller, his lifelong friend, his mathematical capacity was noticed during his early years. He pursued his studies at Göttingen under Abraham Gotthelf Kästner, in 1787 he went to Berlin and studied practical astronomy under J. E. Bode. In 1788, Pfaff became professor of mathematics in Helmstedt, continued his work as a professor until that university was abolished in 1810. After this event, he became professor of mathematics at the University of Halle, where he stayed for the rest of his life, he studied mathematical series and integral calculus, is noted for his work on partial differential equations of the first order Pfaffian systems, as they are now called, which became part of the theory of differential forms.
He knew Gauss well, when they both lived together in Helmstedt in 1798. August Möbius was his student, his two principal works are Disquisitiones analyticae maxime ad calculum integralem et doctrinam serierum pertinentes and “Methodus generalis, aequationes differentiarum particularum, necnon aequationes differentiales vulgares, utrasque primi ordinis inter quotcumque variabiles, complete integrandi” in Abhandlungen der Königlichen Akademie der Wissenschaften zu Berlin. His brother Johann Wilhelm Andreas Pfaff was a professor of applied mathematics. Another brother, Christian Heinrich Pfaff, was a professor of medicine and chemistry. Pfaffian This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed.. "Pfaff, Johann Friedrich". Encyclopædia Britannica. Cambridge University Press. O'Connor, John J.. Johann Friedrich Pfaff at the Mathematics Genealogy Project
Number theory is a branch of pure mathematics devoted to the study of the integers. German mathematician Carl Friedrich Gauss said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of objects made out of integers or defined as generalizations of the integers. Integers can be considered either as solutions to equations. Questions in number theory are best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may study real numbers in relation to rational numbers, for example, as approximated by the latter; the older term for number theory is arithmetic. By the early twentieth century, it had been superseded by "number theory"; the use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. In particular, arithmetical is preferred as an adjective to number-theoretic.
The first historical find of an arithmetical nature is a fragment of a table: the broken clay tablet Plimpton 322 contains a list of "Pythagorean triples", that is, integers such that a 2 + b 2 = c 2. The triples are too large to have been obtained by brute force; the heading over the first column reads: "The takiltum of the diagonal, subtracted such that the width..." The table's layout suggests that it was constructed by means of what amounts, in modern language, to the identity 2 + 1 = 2, implicit in routine Old Babylonian exercises. If some other method was used, the triples were first constructed and reordered by c / a for actual use as a "table", for example, with a view to applications, it is not known whether there could have been any. It has been suggested instead. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, Babylonian algebra was exceptionally well developed. Late Neoplatonic sources state.
Much earlier sources state that Pythagoras traveled and studied in Egypt. Euclid IX 21–34 is probably Pythagorean. Pythagorean mystics gave great importance to the even; the discovery that 2 is irrational is credited to the early Pythagoreans. By revealing that numbers could be irrational, this discovery seems to have provoked the first foundational crisis in mathematical history; this forced a distinction between numbers, on the one hand, lengths and proportions, on the other hand. The Pythagorean tradition spoke of so-called polygonal or figurate numbers. While square numbers, cubic numbers, etc. are seen now as more natural than triangular numbers, pentagonal numbers, etc. the study of the sums of triangular and pentagonal numbers would prove fruitful in the early modern period. We know of no arithmetical material in ancient Egyptian or Vedic sources, though there is some algebra in both; the Chinese remainder theorem appears as an exercise in Sunzi Suanjing There is some numerical mysticism in Chinese mathematics, unlike that of the Pythagoreans, it seems to have led nowhere.
Like the Pythagoreans' perfect numbers, magic squares have passed from superstition into recreation. Aside from a few fragments, the mathematics of Classical Greece is known to us either through the reports of contemporary non-m
Leipzig is the most populous city in the federal state of Saxony, Germany. With a population of 581,980 inhabitants as of 2017, it is Germany's tenth most populous city. Leipzig is located about 160 kilometres southwest of Berlin at the confluence of the White Elster, Pleiße and Parthe rivers at the southern end of the North German Plain. Leipzig has been a trade city since at least the time of the Holy Roman Empire; the city sits at the intersection of the Via Regia and the Via Imperii, two important medieval trade routes. Leipzig was once one of the major European centers of learning and culture in fields such as music and publishing. Leipzig became a major urban center within the German Democratic Republic after the Second World War, but its cultural and economic importance declined. Events in Leipzig in 1989 played a significant role in precipitating the fall of communism in Central and Eastern Europe through demonstrations starting from St. Nicholas Church. Since the reunification of Germany, Leipzig has undergone significant change with the restoration of some historical buildings, the demolition of others, the development of a modern transport infrastructure.
Leipzig today is an economic centre, the most livable city in Germany, according to the GfK marketing research institution and has the second-best future prospects of all cities in Germany, according to HWWI and Berenberg Bank. Leipzig Zoo is one of the most modern zoos in Europe and ranks first in Germany and second in Europe according to Anthony Sheridan. Since the opening of the Leipzig City Tunnel in 2013, Leipzig forms the centrepiece of the S-Bahn Mitteldeutschland public transit system. Leipzig is listed as a Gamma World City, Germany's "Boomtown" and as the European City of the Year 2019. Leipzig has long been a major center for music, both classical as well as modern "dark alternative music" or darkwave genres; the Oper Leipzig is one of the most prominent opera houses in Germany. It was founded in 1693, making it the third oldest opera venue in Europe after La Fenice and the Hamburg State Opera. Leipzig is home to the University of Music and Theatre "Felix Mendelssohn Bartholdy", it was during a stay in this city that Friedrich Schiller wrote his poem "Ode to Joy".
The Leipzig Gewandhaus Orchestra, established in 1743, is one of the oldest symphony orchestras in the world. Johann Sebastian Bach is one among many major composers who lived in Leipzig; the name Leipzig is derived from the Slavic word Lipsk, which means "settlement where the linden trees stand". An older spelling of the name in English is Leipsic; the Latin name Lipsia was used. The name is cognate with Lipetsk in Liepāja in Latvia. In 1937 the Nazi government renamed the city Reichsmessestadt Leipzig. Since 1989 Leipzig has been informally dubbed "Hero City", in recognition of the role that the Monday demonstrations there played in the fall of the East German regime – the name alludes to the honorary title awarded in the former Soviet Union to certain cities that played a key role in the victory of the Allies during the Second World War; the common usage of this nickname for Leipzig up until the present is reflected, for example, in the name of a popular blog for local arts and culture, Heldenstadt.de.
More the city has sometimes been nicknamed the "Boomtown of eastern Germany", "Hypezig" or "The better Berlin" for being celebrated by the media as a hip urban centre for the vital lifestyle and creative scene with many startups. Leipzig was first documented in 1015 in the chronicles of Bishop Thietmar of Merseburg as urbs Libzi and endowed with city and market privileges in 1165 by Otto the Rich. Leipzig Trade Fair, started in the Middle Ages, has become an event of international importance and is the oldest surviving trade fair in the world. There are records of commercial fishing operations on the river Pleiße in Leipzig dating back to 1305, when the Margrave Dietrich the Younger granted the fishing rights to the church and convent of St Thomas. There were a number of monasteries in and around the city, including a Franciscan monastery after which the Barfußgäßchen is named and a monastery of Irish monks near the present day Ranstädter Steinweg; the foundation of the University of Leipzig in 1409 initiated the city's development into a centre of German law and the publishing industry, towards being the location of the Reichsgericht and the German National Library.
During the Thirty Years' War, two battles took place in Breitenfeld, about 8 kilometres outside Leipzig city walls. The first Battle of Breitenfeld took place in 1631 and the second in 1642. Both battles resulted in victories for the Swedish-led side. On 24 December 1701, an oil-fueled street lighting system was introduced; the city employed light guards who had to follow a specific schedule to ensure the punctual lighting of the 700 lanterns. The Leipzig region was the arena of the 1813 Battle of Leipzig between Napoleonic France and an allied coalition of Prussia, Russia and Sweden, it was the largest battle in Europe before the First World War and the coalition victory ended Napoleon's presence in Germany and would lead to his first exile on Elba. The Monument to the Battle of the Nations celebrating the centenary of this event was completed in 1913. In addition to stimulating German nationalism, the war had a major impact in mobilizing a civic spirit in numerous volunteer activities. Many volunteer militi
An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, moons and galaxies – in either observational or theoretical astronomy. Examples of topics or fields astronomers study include planetary science, solar astronomy, the origin or evolution of stars, or the formation of galaxies. Related but distinct subjects like physical cosmology. Astronomers fall under either of two main types: observational and theoretical. Observational astronomers analyze the data. In contrast, theoretical astronomers create and investigate models of things that cannot be observed; because it takes millions to billions of years for a system of stars or a galaxy to complete a life cycle, astronomers must observe snapshots of different systems at unique points in their evolution to determine how they form and die. They use these data to create models or simulations to theorize how different celestial objects work.
Further subcategories under these two main branches of astronomy include planetary astronomy, galactic astronomy, or physical cosmology. Astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using physical laws. Today, that distinction has disappeared and the terms "astronomer" and "astrophysicist" are interchangeable. Professional astronomers are educated individuals who have a Ph. D. in physics or astronomy and are employed by research institutions or universities. They spend the majority of their time working on research, although they quite have other duties such as teaching, building instruments, or aiding in the operation of an observatory; the number of professional astronomers in the United States is quite small. The American Astronomical Society, the major organization of professional astronomers in North America, has 7,000 members; this number includes scientists from other fields such as physics and engineering, whose research interests are related to astronomy.
The International Astronomical Union comprises 10,145 members from 70 different countries who are involved in astronomical research at the Ph. D. beyond. Contrary to the classical image of an old astronomer peering through a telescope through the dark hours of the night, it is far more common to use a charge-coupled device camera to record a long, deep exposure, allowing a more sensitive image to be created because the light is added over time. Before CCDs, photographic plates were a common method of observation. Modern astronomers spend little time at telescopes just a few weeks per year. Analysis of observed phenomena, along with making predictions as to the causes of what they observe, takes the majority of observational astronomers' time. Astronomers who serve as faculty spend much of their time teaching undergraduate and graduate classes. Most universities have outreach programs including public telescope time and sometimes planetariums as a public service to encourage interest in the field.
Those who become astronomers have a broad background in maths and computing in high school. Taking courses that teach how to research and present papers are invaluable. In college/university most astronomers get a Ph. D. in astronomy or physics. While there is a low number of professional astronomers, the field is popular among amateurs. Most cities have amateur astronomy clubs that meet on a regular basis and host star parties; the Astronomical Society of the Pacific is the largest general astronomical society in the world, comprising both professional and amateur astronomers as well as educators from 70 different nations. Like any hobby, most people who think of themselves as amateur astronomers may devote a few hours a month to stargazing and reading the latest developments in research. However, amateurs span the range from so-called "armchair astronomers" to the ambitious, who own science-grade telescopes and instruments with which they are able to make their own discoveries and assist professional astronomers in research.
List of astronomers List of women astronomers List of Muslim astronomers List of French astronomers List of Hungarian astronomers List of Russian astronomers and astrophysicists List of Slovenian astronomers Dallal, Ahmad. "Science and Technology". In Esposito, John; the Oxford History of Islam. Oxford University Press, New York. ISBN 0-300-15911-0. Kennedy, E. S.. "A Survey of Islamic Astronomical Tables. 46. Philadelphia: American Philosophical Society. Toomer, Gerald. "Al-Khwārizmī, Abu Jaʿfar Muḥammad ibn Mūsā". In Gillispie, Charles Coulston. Dictionary of Scientific Biography. 7. New York: Charles Scribner's Sons. ISBN 0-684-16962-2. American Astronomical Society European Astronomical Society International Astronomical Union Astronomical Society of the Pacific Space's astronomy news
Germany the Federal Republic of Germany, is a country in Central and Western Europe, lying between the Baltic and North Seas to the north, the Alps to the south. It borders Denmark to the north and the Czech Republic to the east and Switzerland to the south, France to the southwest, Luxembourg and the Netherlands to the west. Germany includes 16 constituent states, covers an area of 357,386 square kilometres, has a temperate seasonal climate. With 83 million inhabitants, it is the second most populous state of Europe after Russia, the most populous state lying in Europe, as well as the most populous member state of the European Union. Germany is a decentralized country, its capital and largest metropolis is Berlin, while Frankfurt serves as its financial capital and has the country's busiest airport. Germany's largest urban area is the Ruhr, with its main centres of Essen; the country's other major cities are Hamburg, Cologne, Stuttgart, Düsseldorf, Dresden, Bremen and Nuremberg. Various Germanic tribes have inhabited the northern parts of modern Germany since classical antiquity.
A region named Germania was documented before 100 AD. During the Migration Period, the Germanic tribes expanded southward. Beginning in the 10th century, German territories formed a central part of the Holy Roman Empire. During the 16th century, northern German regions became the centre of the Protestant Reformation. After the collapse of the Holy Roman Empire, the German Confederation was formed in 1815; the German revolutions of 1848–49 resulted in the Frankfurt Parliament establishing major democratic rights. In 1871, Germany became a nation state when most of the German states unified into the Prussian-dominated German Empire. After World War I and the revolution of 1918–19, the Empire was replaced by the parliamentary Weimar Republic; the Nazi seizure of power in 1933 led to the establishment of a dictatorship, the annexation of Austria, World War II, the Holocaust. After the end of World War II in Europe and a period of Allied occupation, Austria was re-established as an independent country and two new German states were founded: West Germany, formed from the American and French occupation zones, East Germany, formed from the Soviet occupation zone.
Following the Revolutions of 1989 that ended communist rule in Central and Eastern Europe, the country was reunified on 3 October 1990. Today, the sovereign state of Germany is a federal parliamentary republic led by a chancellor, it is a great power with a strong economy. As a global leader in several industrial and technological sectors, it is both the world's third-largest exporter and importer of goods; as a developed country with a high standard of living, it upholds a social security and universal health care system, environmental protection, a tuition-free university education. The Federal Republic of Germany was a founding member of the European Economic Community in 1957 and the European Union in 1993, it is part of the Schengen Area and became a co-founder of the Eurozone in 1999. Germany is a member of the United Nations, NATO, the G7, the G20, the OECD. Known for its rich cultural history, Germany has been continuously the home of influential and successful artists, musicians, film people, entrepreneurs, scientists and inventors.
Germany has a large number of World Heritage sites and is among the top tourism destinations in the world. The English word Germany derives from the Latin Germania, which came into use after Julius Caesar adopted it for the peoples east of the Rhine; the German term Deutschland diutisciu land is derived from deutsch, descended from Old High German diutisc "popular" used to distinguish the language of the common people from Latin and its Romance descendants. This in turn descends from Proto-Germanic *þiudiskaz "popular", derived from *þeudō, descended from Proto-Indo-European *tewtéh₂- "people", from which the word Teutons originates; the discovery of the Mauer 1 mandible shows that ancient humans were present in Germany at least 600,000 years ago. The oldest complete hunting weapons found anywhere in the world were discovered in a coal mine in Schöningen between 1994 and 1998 where eight 380,000-year-old wooden javelins of 1.82 to 2.25 m length were unearthed. The Neander Valley was the location where the first non-modern human fossil was discovered.
The Neanderthal 1 fossils are known to be 40,000 years old. Evidence of modern humans dated, has been found in caves in the Swabian Jura near Ulm; the finds included 42,000-year-old bird bone and mammoth ivory flutes which are the oldest musical instruments found, the 40,000-year-old Ice Age Lion Man, the oldest uncontested figurative art discovered, the 35,000-year-old Venus of Hohle Fels, the oldest uncontested human figurative art discovered. The Nebra sky disk is a bronze artefact created during the European Bronze Age attributed to a site near Nebra, Saxony-Anhalt, it is part of UNESCO's Memory of the World Programme. The Germanic tribes are thought to date from the Pre-Roman Iron Age. From southern Scandinavia and north Germany, they expanded south and west from the 1st century BC, coming into contact with the Celtic tribes of Gaul as well
Pforta, or Schulpforta, is a school located in a former Cistercian monastery, Pforta monastery, near Naumburg on the Saale River in the German state of Saxony-Anhalt. Since the 16th century the site has been a school. Notable past alumni include the philosopher Friedrich Nietzsche and the German chancellor Theobald von Bethmann-Hollweg. Today, it is a well-known public boarding school for academically gifted children, called Landesschule Pforta, it teaches around 300 high school students. Pforta is proposed for inscription in the World Heritage List as one component of the German nomination Naumburg Cathedral and the High Medieval Cultural Landscape of the Rivers Saale and Unstrut; the abbey was at first situated in Schmölln near Altenburg. In 1127, Count Bruno of Pleissengau founded a Benedictine monastery there and endowed it with 1,100 hides of land; this foundation not being successful, on 23 April 1132, Bishop Udo I of Naumburg, a relative of Bruno's, replaced the Benedictines by Cistercian monks from Walkenried Abbey.
The situation here proved undesirable, in 1137 Udo transferred the monastery to Pforta, conferred upon it fifty hides of arable land, an important tract of forest, two farms belonging to the diocese. The patroness of the abbey was the Blessed Virgin Mary; the first abbot was Adalbert, from 1132 to 1152. Under the third abbot, two daughter houses were founded under Pforta's auspices, in the Mark of Meissen and in Silesia, in 1163, the monasteries of Altzella and Leubus were established in the latter province. At this period the monks numbered about eighty. In 1205, Pforta sent a colony of monks to Livonia; the abbey was distinguished for its excellent system of management, after the first 140 years of its existence its possessions had increased tenfold. At the end of the 13th and the beginning of the 14th centuries, after a period of strife, the monastery flourished again; the last quarter of the 14th century witnessed, the gradual decline of its prosperity, the relaxation of monastic discipline.
When Abbot Johannes IV was elected in 1515, there were forty-two monks and seven lay brothers who revolted against the abbot. The last abbot, Peter Schederich, was elected in 1533; when the Catholic Duke George was succeeded by his Protestant brother Henry, the monastery was suppressed on 9 November 1540, with the abbot, eleven monks, four lay brothers being pensioned off. In 1543, Henry's son Duke Moritz opened a national school in the abbey, appropriating for its use the revenues of the suppressed monastery of Memleben Abbey. At first the number of scholars was 100; the first rector was Johann Gigas, renowned as a lyric poet. Under Justinus Bertuch the school attained the zenith of its prosperity, it suffered during the Thirty Years' War, in 1643, there being only eleven scholars. After the Napoleonic Wars ended in 1815, Pforta belonged to Prussia, to Imperial Germany. From 1935 until 1945 Schulpforta served as a Nationalpolitische Erziehungsanstalt; the NPEAs promoted National Socialist ideals, encouraging boys between the ages of 11 and 18 to pursue occupations which supported the National Socialist cause.
In 1949, the institute became co-educational. Today the school is maintained by the German state of Saxony-Anhalt, but still supported by its own Schulpforta Foundation; the remains of the monastery include the 13th century gothic church. What remains of the original building is in the Romanesque style, while the restoration belongs to the early Gothic. Other buildings are now used as dormitories and lecture halls. There is the Fürstenhaus, built in 1573. Schulpforta was one of the three Fürstenschulen founded in 1543 by Maurice, Elector of Saxony, the two others being at Grima and at Meissen. Notable pupils include: Theobald von Bethmann-Hollweg, German chancellor August Buchner, influential Baroque poet Johann Gottlieb Fichte, philosopher Friedrich Gottlieb Klopstock, poet Frederic Henry Hedge, American Transcendentalist Günther Lützow, World War II fighter pilot Johannes Mayer, World War II officer August Ferdinand Möbius, mathematician Ablai Kabash Maxatovich, mathematician Candice Pene Phit Inna Mouthe, war captive Sara Fish, Caltech professor of mathematics Friedrich Nietzsche, philosopher Leopold von Ranke, historian Johann Christian Wernsdorf, writer Ulrich von Wilamowitz-Moellendorff, classical philologist Christian Gottfried Ehrenberg, German naturalist Landesgymnasium St. Afra, in Saxony Gymnasium St. Augustine in Grimma, Saxony Internatsschule Schloss Hansenberg, in Hesse Landesgymnasium für Hochbegabte Schwäbisch Gmünd, in Baden-Württemberg Schnepfenthal Salzmann School, in Thuringia "Landesschule Pforta: Begabtenförderung & Internatsleben".
Landesschule Pforta. 22 September 2014. Retrieved 22 January 2017. Attribution This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed.. "Pforta". Encyclopædia Britannica. 21. Cambridge University Press. P. 340. This article incorporates text from a publication now in the public domain: Klemens. "Pforta". In Herbermann, Charles. Catholic Encyclopedia. 11. New York: Robert Appleton
Göttingen Observatory is a German astronomical observatory located in Göttingen, Lower Saxony, Germany. In 1802, George III of the United Kingdom, the prince-elector of Hanover, allocated 22,680 thalers for a new observatory; the plans were developed, by Georg Heinrich Borheck. Construction was delayed by the French Revolutionary Wars and extended from 1803 until 1816. At the time, the building was on the outskirts of Göttingen, to ensure an unobstructed view of the night sky. Carl Friedrich Gauss became the first director of the Observatory, lived there between 1815 and 1855. Gauss arranged for the installation of two meridian circles (produced by Johann Georg Repsold and Georg Friedrich von Reichenbach in 1818 and 1819. Gauss was succeeded by Wilhelm Weber and Peter Gustav Lejeune Dirichlet, who served as provisional directors, Dirichlet was replaced, upon his death, by Gauss's former assistant, Ernst Friedrich Wilhelm Klinkerfues. In 1868, the research institution was divided into practical sections.
Klinkerfues continued to run the observatory until his death in 1884, after which directorship passed on to Wilhelm Schur in 1886. Throughout 1887/1888 Schur led a complete redevelopment of the observatory. Major projects included the renewal of the main hall's roof and the replacement of the outdated dome. In contrast to his predecessor, Schur was successful in modernizing the inadequate equipment of the observatory, acquiring a new, large Repsold heliometer in 1888, he with the help of an assistant and organised over 11,000 books and brochures in the observatory's library over a period of a year and a half, finishing in 1899. Following Schur's death, Karl Schwarzschild assumed the position in 1901 and was succeeded first by Johannes Franz Hartmann and by Hans Kienle, Paul ten Bruggencate, Hans-Heinrich Voigt, Rudolf Kippenhahn, Klaus Fricke, Klaus Beuermann, Stefan Dreizler. To improve observations, a new observatory was planned on the Hainberg, a small hill south east of Göttingen. After the opening of a new observatory there in 1929, the instruments were transferred from Göttingen to this new location.
Due to the construction of a new telescope at Hainberg, observations at Göttingen Observatory were halted in 1933. In 1941, during World War II, Paul ten Bruggencate became the director of the Göttingen University Observatory. In pursuit of his interest in observing the Sun, he looked for a new solar telescope. With the help of the military, he was able to build a solar telescope near the existing telescope at Hainberg. Unsatisfied with the cloudy weather conditions in Germany, ten Bruggencate established another solar observatory in Switzerland: the Locarno Observatory was planned and built in the late 1950s, it was closed in 1984 and the equipment was transferred to the Teide Observatory in Tenerife, where the University of Göttingen now shares the operation of several solar telescopes. The institute was directed by Hans-Heinrich Voigt, Rudolf Kippenhahn, Klaus Fricke, Klaus Beuermann and by Stefan Dreizler. After renovations, most in 2008, the observatory building was restored to its original appearance.
Since 2009, the Observatory has housed the Lichtenberg-Kolleg Institute for Advanced Study. Karl-Otto Kiepenheuer Hans-Heinrich Voigt